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The \emph{definite integral} with respect to $x$ of some function $f(x)$ over the closed interval $[a,b]$ is
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The \emph{definite integral} with respect to x of some function $f(x)$ over the closed interval $[a,b]$ is
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defined to be the ``area under the graph of $f(x)$ with respect to $x$'' (if $f(x)$ is negative, then you have a negative area). It is written as:
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defined to be the ``area under the graph of $f(x)$ with respect to x'' (if f(x) is negative, then you have a negative area). It is written as:
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| $$ \int_a^bf(x) \ dx $$ |
$$ \int_a^bf(x) \ dx $$ |
| One way to find the value of the integral is to take a limit of an approximation technique |
One way to find the value of the integral is to take a limit of an approximation technique |
| as the precision increases to infinity. |
as the precision increases to infinity. |
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For example, use a Riemann sum which approximates
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For example, use a Riemann Sum which approximates
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| the area by dividing it into $n$ intervals of equal widths, and then calculating the area |
the area by dividing it into $n$ intervals of equal widths, and then calculating the area |
| of rectangles with the width of the interval and height dependent on the function's value in the interval. |
of rectangles with the width of the interval and height dependent on the function's value in the interval. |
| Let $R_n$ be this approximation, which can be written as |
Let $R_n$ be this approximation, which can be written as |
| $$ R_n = \sum_{i=1}^{n} f(x_i^*) \Delta x $$ |
$$ R_n = \sum_{i=1}^{n} f(x_i^*) \Delta x $$ |
| where $x_i^*$ is some $x$ inside the $i^{\rm th}$ interval. |
where $x_i^*$ is some $x$ inside the $i^{\rm th}$ interval. |
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| Then, the integral would be |
Then, the integral would be |
| $$ \int_a^bf(x) \ dx = \lim_{n \to \infty} R_n = |
$$ \int_a^bf(x) \ dx = \lim_{n \to \infty} R_n = |
| \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x $$ |
\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x $$ |
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| We can use this definition to arrive at some important properties of definite integrals |
We can use this definition to arrive at some important properties of definite integrals |
| ($a$, $b$, $c$ are constant with respect to $x$): |
($a$, $b$, $c$ are constant with respect to $x$): |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| \int_a^bf(x) + g(x) \ dx & = & \int_a^bf(x)\ dx + \int_a^bg(x)\ dx \\ |
\int_a^bf(x) + g(x) \ dx & = & \int_a^bf(x)\ dx + \int_a^bg(x)\ dx \\ |
| \int_a^bf(x) - g(x) \ dx & = & \int_a^bf(x)\ dx - \int_a^bg(x)\ dx \\ |
\int_a^bf(x) - g(x) \ dx & = & \int_a^bf(x)\ dx - \int_a^bg(x)\ dx \\ |
| \int_a^bf(x) \ dx & = & - \int_b^af(x)\ dx \\ |
\int_a^bf(x) \ dx & = & - \int_b^af(x)\ dx \\ |
| \int_a^bf(x) \ dx & = & \int_a^cf(x)\ dx + \int_c^bf(x)\ dx \\ |
\int_a^bf(x) \ dx & = & \int_a^cf(x)\ dx + \int_c^bf(x)\ dx \\ |
| \int_a^bcf(x) \ dx & = & c\int_a^bf(x)\ dx |
\int_a^bcf(x) \ dx & = & c\int_a^bf(x)\ dx |
| \end{eqnarray*} |
\end{eqnarray*} |
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| There are other generalisations about integrals, but many require the Fundamental Theorem of Calculus. |
There are other generalisations about integrals, but many require the Fundamental Theorem of Calculus. |
